Exponential growth of torsion in abelian coverings

Raimbault, Jean¹

¹Institut de mathématiques de Jussieu, 4, place Jussieu, Case 247, 75252 Paris Cédex 5, France

Algebraic and Geometric Topology, Tome 12 (2012) no. 3, pp. 1331-1372

Résumé

We show exponential growth of torsion numbers for links whose first nonzero Alexander polynomial has positive logarithmic Mahler measure. This extends a theorem of Silver and Williams to the case of a null first Alexander polynomial and provides a partial solution for a conjecture of theirs.

Détail

DOI : 10.2140/agt.2012.12.1331

Keywords: Reidemeister torsion, $\ell^2$–torsion

Bibliographie

[1] N Bergeron, A Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, to appear in J. Inst. Math. Jussieu

[2] J C Cha, S Friedl, T Kim, The cobordism group of homology cylinders, Compos. Math. 147 (2011) 914

[3] B Clair, K Whyte, Growth of Betti numbers, Topology 42 (2003) 1125

[4] G Everest, T Ward, Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer (1999)

[5] F González-Acuña, H Short, Cyclic branched coverings of knots and homology spheres, Rev. Mat. Univ. Complut. Madrid 4 (1991) 97

[6] C M Gordon, Knots whose branched cyclic coverings have periodic homology, Trans. Amer. Math. Soc. 168 (1972) 357

[7] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[8] J Hempel, 3–Manifolds, Ann. of Math. Studies 86, Princeton Univ. Press (1976)

[9] E Kowalski, The large sieve and its applications, Cambridge Tracts in Mathematics 175, Cambridge Univ. Press (2008)

[10] M Lackenby, Finite covering spaces of 3–manifolds, from: "Proceedings of the International Congress of Mathematicians Volume II", Hindustan Book Agency (2010) 1042

[11] T Y Lam, Serre's problem on projective modules, Springer Monographs in Mathematics, Springer (2006)

[12] W M Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983) 356

[13] T Le, Homology torsion growth and Mahler measure

[14] W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer (1997)

[15] P Linnell, W Lück, R Sauer, The limit of $\mathbb{F}_p$–Betti numbers of a tower of finite covers with amenable fundamental groups, Proc. Amer. Math. Soc. 139 (2011) 421

[16] W Lück, Approximating $L^2$–invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994) 455

[17] W Lück, $L^2$–invariants: theory and applications to geometry and $K$–theory, Ergeb. Math. Grenzgeb. 44, Springer (2002)

[18] J P Mayberry, K Murasugi, Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982) 143

[19] J Porti, Mayberry–Murasugi's formula for links in homology 3–spheres, Proc. Amer. Math. Soc. 132 (2004) 3423

[20] R Riley, Growth of order of homology of cyclic branched covers of knots, Bull. London Math. Soc. 22 (1990) 287

[21] A Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications 77, Cambridge Univ. Press (2000)

[22] K Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics 128, Birkhäuser (1995)

[23] J P Serre, Local algebra, Springer Monographs in Mathematics, Springer (2000)

[24] D S Silver, S G Williams, Mahler measure, links and homology growth, Topology 41 (2002) 979

[25] D S Silver, S G Williams, Torsion numbers of augmented groups with applications to knots and links, Enseign. Math. 48 (2002) 317

[26] D S Silver, S G Williams, Dynamics of twisted Alexander invariants, Topology Appl. 156 (2009) 2795

[27] V Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser (2001)